Chessboard PuzzlesPart 1: Domination

Dan Freeman

February 20, 2014

Villanova UniversityMAT 9000 Graduate Math Seminar

Brief Overview of Chess• Chess is a classic board game that has

been played for at least 1,200 years• Chess is a 2-player turn-based game

played on an 8x8 board• There are six different types of pieces:

pawn, knight, bishop, rook, queen and king• The objective of the game is to put the

opponent’s king in a position in which it cannot escape attack; this position is known as checkmate

Chess Starting Positions

King

Queen

Rook

Bishop

Knight

Pawn

Domination Defined• A dominating set of chess pieces is one

such that every square on the chessboard is either occupied by a piece in the set or under attack by a piece in the set

• The domination number for a certain piece and certain size chessboard is the minimum number of such pieces required to “dominate” or “cover” the board

• Domination numbers are denoted by γ(Pmxn) where P represents the type of chess piece (see legend to the right) and m and n are the number of rows and columns of the board, respectively

King – K

Queen – Q

Rook – R

Bishop – B

Knight – N

Domination Number Notation

Rook Movement• Rooks move horizontally and vertically• Rooks are allowed to move any number of

squares in one direction as long as they do not take the place of a friendly piece or pass through any piece (own or opponent’s) currently on the board

• In the example below, the white rook can move to any of the squares with a white circle and the black rook can move to any of the squares with a black circle

Rooks Domination• Domination among rooks is the simplest of

all chess pieces• For a square nxn chessboard, the rooks

domination number is simply n• For a general rectangular mxn chessboard,

γ(Rmxn) = min(m, n)

Proof that γ(Rnxn) = n• Two Russian brothers Akiva and Isaak

Yaglom proved this:– First, suppose there are fewer than n rooks

placed on an nxn board. Then there must be at least one row and at least one column that contain no rooks. Hence, the square where this empty row and column intersect is uncovered. Thus, γ(Rnxn) ≥ n.

– Second, if n rooks are placed along a single row or down a single column, the entire board is clearly dominated. That is, γ(Rnxn) ≤ n.

– Lastly, since γ(Rnxn) ≥ n and γ(Rnxn) ≤ n, we conclude that γ(Rnxn) = n.

• The fact that γ(Rmxn) = min(m, n) follows immediately from the above

Bishop Movement• Bishops move diagonally• Bishops are allowed to move any number

of squares in one direction as long as they do not take the place of a friendly piece or pass through any piece (own or opponent’s) currently on the board

• In the example below, the white bishop can move to any of the squares with a white circle and the black bishop can move to any of the squares with a black circle

Bishops Domination• As with rooks, γ(Bnxn) = n (though in

general, γ(Bmxn) ≠ min(m, n))

• The proof that γ(Bnxn) = n for bishops is more involved than that for rooks

• The proof starts with rotating the chessboard 45 degrees, as shown below

45°

5x4 Black Square

Proof that γ(Bnxn) = n• Yaglom and Yaglom proved this:

– Suppose n = 8 (the following argument works for all even n).

– Clearly, from rotating the board 45 degrees, we see a 5x4 construction of dark (black) squares in the middle of the board. Therefore, at least 4 bishops are needed to cover all of the black squares. By symmetry, at least 4 bishops are needed to cover all of the light (white) squares. Therefore, γ(B8x8) ≥ 4 + 4 = 8.

– On the other hand, if we place 8 bishops in the fourth column of a chessboard, we find that the entire board is covered. Thus, γ(B8x8) ≤ 8.

– Since γ(B8x8) ≥ 8 and γ(B8x8) ≤ 8, it follows that γ(B8x8) = 8 and for general even n, γ(Bnxn) = n.

Bishops Domination on 8x8 Board

Proof that γ(Bnxn) = n– Now suppose n is odd and let n = 2k + 1.– The board corresponding to squares of one color

(without loss of generality, suppose this color is white) will contain a (k + 1)x(k + 1) group of squares; hence, at least k + 1 bishops are needed to cover the white squares.

– Likewise, the board corresponding to black squares will contain a kxk group of squares and hence at least k bishops are needed to cover the black squares.

– Thus, at least (k + 1) + k = 2k + 1 = n bishops are needed to dominate the entire nxn board.

– To see that γ(Bnxn) ≤ n, observe that if n bishops are placed down the center column, the entire board is covered.

– In conclusion, γ(Bnxn) = for all n.

King Movement• Kings are allowed to move exactly one

square in any direction as long as they do not take the place of a friendly piece

• In the example below, the king can move to any of the squares with a white circle

Kings Domination• Shown are examples of

9 kings dominating 7x7, 8x8 and 9x9 boards

• For 7 ≤ n ≤ 9, γ(Knxn) = 9

Kings Domination• No matter where one

places a king on any of the 7x7, 8x8 or 9x9 boards, only one of the nine dark orange squares will be covered

Kings Domination• Thus, 32 = 9 is the domination number for

square chessboards where n = 7, 8 or 9• This triplet pattern continues for larger

boards:– For n = 10, 11 and 12, γ(Knxn) = 42 = 16.

– For n = 13, 14 and 15, γ(Knxn) = 52 = 25.

• Thus, the general formula for the kings domination number can be written making use of the greatest integer or floor function:

– γ(Knxn) = └(n + 2) / 3┘2.

• Generalizing even further, the formula for rectangular boards is:

– γ(Kmxn) = └(m + 2) / 3┘* └(n + 2) / 3┘.

Knight Movement• Knights move two squares in one direction

(either horizontally or vertically) and one square in the other direction as long as they do not take the place of a friendly piece

• Knights’ moves resemble an L shape• Knights are the only pieces that are allowed

to jump over other pieces• In the example below, the white and black

knights can move to squares with circles of the corresponding color

Knights Domination• No explicit formula is known for

the knights domination number• However, several values of

γ(Nnxn) have been verified– The first 20 knights domination

numbers appear in the table to the right.

• As can be seen from the table, as n increases, γ(Nnxn) increases in no discernible pattern

n γ(Nnxn) 1 12 43 44 45 56 87 108 129 14

10 1611 2112 2413 2814 3215 3616 4017 4618 5219 5720 62

Knights Domination

Queen Movement• Queens move horizontally, vertically and

diagonally• Queens are allowed to move any number

of squares in one direction as long as they do not take the place of a friendly piece or pass through any piece (own or opponent’s) currently on the board

• In the example below, the queen can move to any of the squares with a black circle

Queens Domination• Domination among queens is the most

complicated and interesting of all chess pieces, as well as the least understood

• No formula is known for the queens domination number, but lower and upper bounds have been established

Arrangement of 5 Queens

Dominating 8x8 Board

Queens Diagonal Domination• However, a formula for the queens diagonal

domination number is known• The queens diagonal domination number,

denoted diag(Qnxn), is the minimum number of queens all placed along the main diagonal required to cover the board

• For all n, diag(Qnxn) = n – max(|mid-point free, all even or all odd, subset of {1, 2, 3, …, n}|)

• A mid-point free set is a set in which for any given pair of elements in the set, the midpoint of those two numbers is not in the set

Upper and Lower Bounds• Upper bound for queens domination

number: Welch proved that for n = 3m + r, 0 ≤ r < 3, γ(Qnxn) ≤ 2m + r

• Lower bound for queens domination number: Spencer proved that for any n, γ(Qnxn) ≥ ½*(n – 1)

• Weakley improved on Spencer’s lower bound by showing that if the lower bound is attained, that is, if γ(Qnxn) = ½*(n – 1), then n ≡ 3 mod 4

• Corollaries of Spencer’s improved lower bound include:– γ(Q7x7) = 4.

– For n = 4k + 1, γ(Qnxn) ≥ ½*(n + 1) = 2k + 1.

Queens Domination Numbers• The lower bounds on the previous slide

allow us to narrow the number of possibilities considerably for the queens domination number for larger chess boards, as shown in the table below

n γ(Qnxn) 14 7 or 815 7, 8 or 916 8 or 917 918 919 9 or 1020 10 or 1121 1122 11 or 1223 11, 12 or 1324 12 or 1325 13

Sources Cited• J.J. Watkins. Across the Board: The

Mathematics of Chessboard Problems. Princeton, New Jersey: Princeton University Press, 2004.

• J. Nunn. Learn Chess. London, England: Gambit Publications, 2000.

• “Chess.” Wikipedia, Wikimedia Foundation. http://en.wikipedia.org/wiki/Chess

• “A006075 – OEIS.” http://oeis.org/A006075